Fundamentals

1
Fundamentals

• What kind of data do computers work with?
• Deep down inside, its all 1s and 0s
• What can you do with 1s and 0s?
• Boolean algebra operations
• These operations map directly to hardware circuits

2
Computers are binary devices

• Computers use voltages to represent information
• Voltages are usually limited to 2.5-5.0V to
  minimize power consumption
• This is only enough to represent two discrete
  (digital) signals
• Low and High
• 0 and 1
• False and True
• Why?
• To account for noise
• To prevent transitional errors
• How can we use these two signals to represent
  numbers?

Good!
Bad!
3
Decimal review

• Numbers consist of a bunch of digits, each with a
  weight
• These weights are all powers of the base, which
  is 10. We can rewrite this
• To find the decimal value of a number, multiply
  each digit by its weight and sum the products.

(1 x 102) (6 x 101) (2 x 100) (3 x 10-1)
(7 x 10-2) (5 x 10-3) 162.375
4
Converting binary to decimal

• We can use the same trick to convert binary, or
  base 2, numbers to decimal. This time, the
  weights are powers of 2.
• Example 1101.01 in binary
• The decimal value is

(1 x 23) (1 x 22) (0 x 21) (1 x 20) (0 x
2-1) (1 x 2-2) 8 4 0
1 0 0.25 13.25
5
Opposite converting decimal to binary

• To convert an integer, keep dividing by 2 until
  the quotient is 0. Collect the remainders in
  reverse order.
• To convert a fraction, keep multiplying the
  fractional part by 2 until it becomes 0. Collect
  the integer parts (in forward order).
• This may not terminate!
• Example 162.375
• So, 162.37510 10100010.0112

162 / 2 81 rem 0 81 / 2 40 rem 1 40 / 2
20 rem 0 20 / 2 10 rem 0 10 / 2 5 rem 0
5 / 2 2 rem 1 2 / 2 1 rem 0 1 / 2
0 rem 1
0.375 x 2 0.750 0.750 x 2 1.500 0.500 x 2
1.000
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Why does this work?

• This works for converting from decimal to any
  base
• Why? Think about converting 162.375 from decimal
  to decimal.
• Each division strips off the rightmost digit
  (the remainder). The quotient represents the
  remaining digits in the number.
• Similarly, each multiplication strips off the
  leftmost digit (the integer part). The fraction
  represents the remaining digits.

162 / 10 16 rem 2 16 / 10 1 rem 6 1 /
10 0 rem 1
0.375 x 10 3.750 0.750 x 10 7.500 0.500 x 10
5.000
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Base 8 and base 16 are useful too

• Octal (base 8) digits range from 0 to 7. Since 8
  23, one octal digit is equivalent to 3 binary
  digits.
• Hexadecimal (base 16) digits are 0, 1, 2, 3, 4,
  5, 6, 7, 8, 9, A, B, C, D, E, and F. Since 16
  24, one hex digit is equivalent to 4 binary
  digits.
• We typically use octal and hex as a shorthand for
  those long messy binary numbers.

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Binary, octal, hexadecimal

• Converting from octal (or hex) to binary is easy
  just replace each octal (or hex) digit with the
  equivalent 3 (or 4) bit sequence
• To convert from binary to octal (or hex)
  starting from the binary point, make groups of 3
  (or 4) bits. Add 0s to the ends of the number if
  needed. Then, just convert each bit group to its
  corresponding octal (or hex) digit.

261.358 2 6 1 . 3 58 010 110 001 . 011 1012
261.3516 2 6 1 . 3 516 0010 0110 0001 . 00
11 01012
These binary numbers are not equivalent!
10110100.0010112 010 110 100 . 001 0112 2 6 4
. 1 38
These numbers are equivalent!
10110100.0010112 1011 0100 . 0010 11002 B 4 .
2 C16
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Functions

• A computer takes input and produces output...
  just like a math function!
• We can express math functions in two ways
• We can represent binary functions in two ways
  too
• As a Boolean expression
• As a truth table. It shows all possibilities for
  an expression.

As an expression
As a function table
f(x,y) 2x y x x y 2(x y/2) ...
10
Basic Boolean operations

• There are three basic operations. Each can be
  implemented in hardware using a primitive logic
  gate

NOT (complement) on one input
AND (product) of two inputs
OR (sum) of two inputs
Operation
Expression
xy
x y
x
Truth table
Logic gate
11
Boolean expressions

• We can use these basic operations to form more
  complex expressions and equations
• Some terminology and notation
• f is the name of the function
• (x,y,z) are the input variables. Each variable
  represents a 1 or 0. Listing input variables is
  optional instead we often write
• A literal is an occurrence of an input variable
  or its complement. The function above has four
  literals x, y, z, and x.
• Without parentheses, the NOT operation has the
  highest precedence, followed by AND, and finally
  OR. Fully parenthesized, this function is

f(x,y,z) (x y)z x
f (x y)z x
f(x,y,z) (((x (y))z) x)
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Larger circuits

• We can build circuits for arbitrary expressions,
  using our basic gates

f(x,y,z) (x y)z x
(x y)z
y
(x y)
x
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Truth tables

• We can make a truth table too, by evaluating the
  function for all possible inputs.
• In general, a function with n inputs will have 2n
  possible inputs.
• The inputs are typically listed in binary order.

f(x,y,z) (x y)z x
f(0,0,0) (0 1)0 1 1 f(0,0,1) (0 1)1
1 1 f(0,1,0) (0 0)0 1 1 f(0,1,1) (0
0)1 1 1 f(1,0,0) (1 1)0 0
0 f(1,0,1) (1 1)1 0 1 f(1,1,0) (1 0)0
0 0 f(1,1,1) (1 0)1 0 1
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Summary

• Summary
• To minimize analog voltage problems, computers
  are binary devices
• That means we have to think in terms of base 2
• The basic operations on binary values include
  AND, OR, and NOT
• These can be directly implemented in hardware
  (eventually well show how to do useful stuff
  with these operations)
• Expressions and truth tables are used to design
  and describe circuits
• Weve already seen some of the recurring themes
  of architecture
• We use 0 and 1 as abstractions for analog
  voltages
• Gates are also an abstraction of the underlying
  technology
• We showed how to represent numbers using just
  these two signals
• Next time
• Using Boolean algebra to simplify expressions
• This in turn will yield a simpler circuit